تقرير
Decompositions of high-frequency Helmholtz solutions via functional calculus, and application to the finite element method
| العنوان: | Decompositions of high-frequency Helmholtz solutions via functional calculus, and application to the finite element method |
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| المؤلفون: | Galkowski, Jeffrey, Lafontaine, David, Spence, Euan A., Wunsch, Jared |
| سنة النشر: | 2021 |
| المجموعة: | Computer Science Mathematics |
| مصطلحات موضوعية: | Mathematics - Analysis of PDEs, Mathematics - Numerical Analysis |
| الوصف: | Over the last ten years, results from [Melenk-Sauter, 2010], [Melenk-Sauter, 2011], [Esterhazy-Melenk, 2012], and [Melenk-Parsania-Sauter, 2013] decomposing high-frequency Helmholtz solutions into "low"- and "high"-frequency components have had a large impact in the numerical analysis of the Helmholtz equation. These results have been proved for the constant-coefficient Helmholtz equation in either the exterior of a Dirichlet obstacle or an interior domain with an impedance boundary condition. Using the Helffer-Sj\"ostrand functional calculus, this paper proves analogous decompositions for scattering problems fitting into the black-box scattering framework of Sj\"ostrand-Zworski, thus covering Helmholtz problems with variable coefficients, impenetrable obstacles, and penetrable obstacles all at once. These results allow us to prove new frequency-explicit convergence results for (i) the $hp$-finite-element method applied to the variable coefficient Helmholtz equation in the exterior of a Dirichlet obstacle, when the obstacle and coefficients are analytic, and (ii) the $h$-finite-element method applied to the Helmholtz penetrable-obstacle transmission problem. In particular, the result in (i) shows that the $hp$-FEM applied to this problem does not suffer from the pollution effect. |
| نوع الوثيقة: | Working Paper |
| URL الوصول: | http://arxiv.org/abs/2102.13081 |
| رقم الأكسشن: | edsarx.2102.13081 |
| قاعدة البيانات: | arXiv |
| الوصف غير متاح. |